Lecture 13, October 22
• Polar coordinates. Expressing a double integral in polar coordinates, one has
f (x, y) dA =
f (r cos θ, r sin θ) · rdr dθ
R
for some suitable limits of integration that describe the region R.
.....................................................................................
Example 1. If R is the region depicted on the left side of the figure, then
1 √1−x2
(x2 + y 2 ) dA =
(x2 + y 2 ) dy dx.
0
R
0
Expressing this integral in polar coordinates, one can also write it as
π/2 1
2
2
(x + y ) dA =
r3 dr dθ.
0
R
0
Example 2. We use polar coordinates in order to compute the integral
√2 √4−y2 I=
x2 + y 2 dx dy.
0
y
In this case, the
region of integration is bounded by the line x = y on the left and by
the circle x = 4 − y 2 on the right. Note that these two intersect when
y = 4 − y 2 =⇒ y 2 = 4 − y 2 =⇒ 2y 2 = 4 =⇒ y 2 = 2.
√
This explains the upper limit of integration y = 2. The region of integration is thus
the one depicted on the right and we can describe it using polar coordinates to get
π/4 3 2
π/4
π/4 2
r
2π
8
r · r dr dθ =
dθ =
dθ =
.
I=
3 0
3
3
0
0
0
0
1
y=x
x2 + y 2 = 1
√
2
x2 + y 2 = 4
1
2
Figure: The regions of integration for Examples 1 and 2.